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A333322
Decimal expansion of (3/8) * sqrt(3).
1
6, 4, 9, 5, 1, 9, 0, 5, 2, 8, 3, 8, 3, 2, 8, 9, 8, 5, 0, 7, 2, 7, 9, 2, 3, 7, 8, 0, 6, 4, 7, 0, 2, 1, 3, 7, 6, 0, 3, 5, 5, 1, 9, 7, 0, 1, 7, 8, 8, 9, 2, 7, 3, 5, 5, 2, 0, 9, 2, 7, 6, 1, 7, 2, 9, 4, 4, 7, 4, 8, 8, 1, 3, 4, 0, 8, 0, 0, 0, 1, 3, 9, 0, 5, 4, 2, 9, 8, 2, 0, 0, 3, 3, 9, 6, 8, 2, 1, 5, 8, 7, 8, 3, 5, 9, 8, 0, 3, 0, 3, 0, 7, 7, 7, 5, 1, 3, 6, 3, 6
OFFSET
0,1
COMMENTS
This is the area of the regular hexagon of diameter 1.
From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)
For any triangle ABC, where (A,B,C) are the angles:
sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],
cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],
and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:
(ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).
The equalities are obtained only when triangle ABC is equilateral. (End)
REFERENCES
O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.
D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.
LINKS
Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.
FORMULA
Equals A104954/2 or A104956/4.
EXAMPLE
0.649519052838328985...
MATHEMATICA
RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]
PROG
(PARI) sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022
CROSSREFS
Cf. A002194 (sqrt(3)), A104954.
Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).
Sequence in context: A231535 A019931 A195414 * A348896 A153630 A113276
KEYWORD
cons,nonn
AUTHOR
Kritsada Moomuang, Mar 15 2020
STATUS
approved